Find the error in finding the following limit: \lim_(\theta \to 0) (1 - cos\theta)/(2sin^2\theta)

Question

Find the error in finding the following limit:


`\lim_(\theta \to 0) (1 - cos\theta)/(2sin^2\theta)`

Answer 1

`\displaystyle \lim_(\theta \to 0) (1-\cos\theta)/(2\sin^2\theta)=(1)/(4)`

Explanation:

The error is in line 3 of the given calculations.

At this point, do not expand the brackets of the denominator. Instead, factor 2 out of the one of the brackets, then cancel the common factor sin²θ.

`\begin{aligned}\displaystyle \lim_(\theta \to 0) (1-\cos\theta)/(2\sin^2\theta)&=\lim_(\theta \to 0) \left((1-\cos\theta)/(2\sin^2\theta)\right)\cdot\left((1+\cos\theta)/(1+\cos\theta)\right)\\\\&=\lim_(\theta \to 0)(1-\cos^2\theta)/((2\sin^2\theta)(1+\cos\theta))\\\\&=\lim_(\theta \to 0)(\sin^2\theta)/(2(\sin^2\theta)(1+\cos\theta))\\\\&=\lim_(\theta \to 0)(1)/(2(1+\cos\theta))\\\\&=(1)/(2)\cdot\lim_(\theta \to 0)(1)/(1+\cos\theta)\\\\\end{aligned}`

As θ → 0:

`\begin{aligned}&=(1)/(2)\cdot(1)/(1+\cos(0))\\\\&=(1)/(2)\cdot(1)/(1+1)\\\\&=(1)/(2)\cdot(1)/(2)\\\\&=(1)/(4)\end{aligned}`